A092976 Consider all partitions of n into parts all of which are divisors of n; a(n) = number of distinct values taken by the product of the parts.

1, 1, 2, 2, 3, 2, 7, 2, 5, 4, 10, 2, 19, 2, 13, 13, 9, 2, 37, 2, 29, 17, 19, 2, 61, 6, 22, 10, 39, 2, 247, 2, 17, 25, 28, 25, 127, 2, 31, 29, 97, 2, 450, 2, 59, 82, 37, 2, 217, 8, 146, 37, 69, 2, 271, 37, 133, 41, 46, 2, 1558, 2, 49, 112, 33, 43, 1038, 2, 89, 49, 1105, 2, 469, 2, 58, 211, 99, 49, 1423, 2, 353

Offset: 0

Author

Amarnath Murthy, Mar 27 2004

Keywords

• nonn

Comments

a(n) > tau(n) + A(n) + R(n), where tau(n) = number of divisors of n, A(n) = product of powers of nontrivial divisors whose sum with multiplicity is < n and R(n) = numbers of the form r^(k) > n where r is a divisor of n and k <= n/r.

Examples

			a(10) = 10, the numbers arising are 1,2,4,5,8,10,16,20,25 and 32; e.g. 25 = 5*5, 8 = 2*2*2*1*1*1*1, 32 = 2*2*2*2*2, etc.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A000040, A092975.

Programs

Maple

with(numtheory):
a:= proc(n) local b, l, s;
      l:= sort([divisors(n)[]]);
      b:= proc(n,i,p)
            if n<0 then
          elif n=0 then s:= s union {p}
          elif i=0 then
          else b(n-l[i], i, p*l[i]); b(n, i-1, p)
            fi
          end;
      s:= {};
      b(n, nops(l), 1);
      nops(s)
    end:
seq(a(n), n=0..60);  # Alois P. Heinz, Feb 21 2011

Mathematica

a[n_] := Module[{ b, l, s}, l = Divisors[n]; b[m_, i_, p_] := Which[m<0, , m == 0, s = Union[s, {p}], i == 0, , True, b[m - l[[i]], i, p*l[[i]]]; b[m, i-1, p]]; s = {}; b[n, Length[l], 1]; Length[s]]; Table[a[n], {n, 1, 60}] (* Jean-François Alcover, Nov 11 2015, after Alois P. Heinz *)

Formula

a(n) = 2 <=> n in { A000040 }. - Alois P. Heinz, Oct 28 2021

Extensions

More terms from Vladeta Jovovic, Aug 06 2005

More terms from Alois P. Heinz, Feb 21 2011

a(0)=1 prepended by Alois P. Heinz, Oct 28 2021